Why does pyproj.Proj forward projection not seem to account for lat lon origin?

Question

I am getting confused about how projections using pyproj.Proj are defined with respect to a point of tangency / lat lon origin.

Consider the following code:

import pyproj

p = pyproj.Proj('+proj=tmerc +lat_0=55 +lon_0=-1 +a=6378137 +b=6356752.3 +units=m +no_defs')
x, y = p(55, -1)

Now given that I specified the origin for latitude and longitude, I would expect that when specifying those coordinates I would be able to assert x == 0 and y == 0, however I actually get (7571700.820174289, -6296411.725576388).

Can anyone explain why this is the case? My knowledge of projection/coordinate systems is limited, but I did my best to understand PROJ Cartographic help and a related wikibook page.

Many thanks in advance to any who can help set me straight and put me in the right direction :-)

EDIT: Update 1

Thanks to @lusitanica and their helpful answer, I have now tried setting the scale factor to 1 and rerunning:

x, y = pyproj.Proj('+proj=tmerc +lat_0=55 +lon_0=-1 +k_0=1 +a=6378137 +b=6356752.3 +units=m +no_defs', preserve_units=True)(55, -1)

Unfortunately this gives (7571700.820174289, -6296411.725576388) as before, so the question is what other information is needed for the projection string?

LuisTavares LuisTavares · Accepted Answer · 2019-06-16T18:58:55

I can confirm independently that the result should be (0,0)

from math import *

# GRS-80
a = 6378137.
equad =0.00669437999

# Natural Origin 
lat0=55.
lon0=-1.

############################################################################
# Meridian Arc
############################################################################
def arcmer(a,equad,lat1,lat2):
    b=a*sqrt(1-equad)
    n=(a-b)/(a+b)
    a0=1.+((n**2)/4.)+((n**4)/64.)
    a2=(3./2.)*(n-((n**3)/8.))
    a4=(15./16.)*((n**2)-((n**4)/4.))
    a6=(35./48.)*(n**3)

    s1=a/(1+n)*(a0*lat1-a2*sin(2.*lat1)+a4*sin(4.*lat1)-a6*sin(6.*lat1))
    s2=a/(1+n)*(a0*lat2-a2*sin(2.*lat2)+a4*sin(4.*lat2)-a6*sin(6.*lat2))
    return s2-s1
#############################################################################
# Direct projection Gauss-Kruger
#############################################################################
def geogauss(lat,lon,a,equad,lat0,lon0):

    lat0=radians(lat0)
    lon0=radians(lon0)

    lat=radians(lat)
    lon=radians(lon)

    lon=lon-lon0

    N=a/sqrt(1-equad*(sin(lat))**2)
    RO=a*(1-equad)/((1-equad*(sin(lat)**2))**(3./2.))

    k1=(N/RO)+(4.*(N**2)/(RO**2))-((tan(lat))**2)

    k2=(N/RO)-((tan(lat))**2)

    k3=N/RO*(14.-58.*((tan(lat)))**2)+40.*((tan(lat))**2)+((tan(lat))**4)-9.

    x=lon*N*cos(lat)+(lon**3)/6.*N*((cos(lat))**3)*k2+(lon**5)/120.*N*((cos(lat))**5)*k3

    y=arcmer(a,equad,lat0,lat)+(lon**2)/2.*N*sin(lat)*cos(lat)+((lon**4)/24.)*N*sin(lat)*((cos(lat))**3)*k1

    return x,y

lat=55.
lon=-1.
coordinates = geogauss(lat,lon,a,equad,lat0,lon0) 
print lat,lon
print "x= %.3f" %coordinates[0]
print "y= %.3f" %coordinates[1]

Result:

55.0 -1.0

x= 0.000

y= 0.000

In my code I'm considering a Scale Factor at Natural Origin of 1.

In your PROJ string I can't see any. Try to set the Scale factor to +k_0=1

If that doesn't help there's something else missing from your PROJ string because the result is realy (0,0).

Unless there is a bug with PyProj, which I doubt.

Why does pyproj.Proj forward projection not seem to account for lat lon origin?

EDIT: Update 1

2 Answers